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On the variation of Teichmüller's metric

Published online by Cambridge University Press:  14 November 2011

Frederick P. Gardiner
Frederick P. Gardiner
Affiliation:
Brooklyn College, CUNY, Brooklyn, N.Y.
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Synopsis

The main result of this article is the calculation of the first derivative of Teichmüller's metric from an inequality of Reich and Strebel. Furthermore, from the same inequality one is able to calculate information about the difference quotient for the second derivative. From the techniques used here it does not seem possible to determine whether the metric is C2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 143 - 152
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Ahlfors, L. V.. On quasiconformal mappings. J. Analyse Math. 3 (1954), 1–58.CrossRefGoogle Scholar
2Ahlfors, L. V.. Lectures on Quasiconformal Mappings. (Princeton, N.J.: Van Nostrana, 1966).Google Scholar
3Ahlfors, L. V. and Bers, L.. Riemann's mapping theorem for variable metrics. Ann. of Math. 72 (1960), 385–404.CrossRefGoogle Scholar
4Bers, L.. Quasiconformal mappings and Teichmüller's theorem. In Analytic Functions. 89–119, by Nevanlinna, R.et al. (Princeton, N.J.: Princeton Univ. Press, 1960).Google Scholar
5Bers, L.A non-standard integral equation with applications to quasiconformal mappings. Acta Math. 116 (1966), 113–134.CrossRefGoogle Scholar
6Bers, L. A new proof of a fundamental inequality for quasiconformal mappings, to appear, (1979).CrossRefGoogle Scholar
7Earle, C. J.. The Teichmüller distance differentiable. Duke Math. J. 44 (1977), 389–397.CrossRefGoogle Scholar
8Federer, H.. Geometric Measure Theory (New York: Springer, 1969).Google Scholar
9Reich, E.. On the decomposition of a class of plane quasiconformal mappings. Comment. Math. Helv. 53 (1978), 15–27.CrossRefGoogle Scholar
10Reich, E.. A generalized Dirichlet integral. J. Analyse Math. 30 (1976), 456–463.CrossRefGoogle Scholar
11Reich, E. and Strebel, K.. Extremal quasiconformal mappings with given boundary values. In Contributions to Analysis, 375–392, ed. Ahlfors, L. V.et al. (New York and London: Academic Press, 1974).CrossRefGoogle Scholar
12Reich, E.. Teichmüller mappings which keep the boundary points fixed. Trans. Amer. Math. Soc. 138 (1969), 211–222.CrossRefGoogle Scholar
13Strebel, K.. On the trajectory structure of quadratic differentials. Discontinuous Groups and Riemann Surfaces. Ann. of Math. Studies No. 79, 419–438 (Princeton, N.J.: Princeton Univ. Press, 1974).CrossRefGoogle Scholar
14Strebel, K.. On the existence of extremal Teichmüller mappings. J. Analyse Math. 30 (1976), 464–480.CrossRefGoogle Scholar
15Strebel, K.. On quasiconformal mappings of open Riemann surfaces. Comment. Math. Helv. 53 (1978), 301–321.CrossRefGoogle Scholar